This happens when the temperature gradient across the solid changes with time.
This may be due to unstable boundary temperatures at startup, sudden temperature fluctuations during steady state conditions, or internal generation of heat.
Startup of a Furnace
Heat Treatment of Solids
Deep Oil Frying
Change of Weather
Unidirectional Unsteady State Case x y z x x+ x q in q out A = V = Solid properties: , c P Mass = Heat Balance Across x: q in - q out = Rate of heat accumulation Rate of heat accumulation = Using Fourier’s Law: q in = q out = c P y z x y z x y z x y z T t - k y z T x x - k y z T x x + x
Unidirectional Unsteady State Case The Heat Balance becomes: - = Simplifying: - x From Calculus: Final Equation: but - k y z T x x - k y z T x x + x x y z c P T t T t = k c P T x x + x T x x T x x + x - T x x x = 2 T x 2 T t = k c P = 2 T x 2
A slab of meat 25.4 mm thick originally at a uniform temperature of 10 o C is to be cooked from both sides until the center reaches 121 o C in an oven at 177 o C. The convection coefficient can be assumed constant at 25.6 W/m 2 -K. Neglect any latent heat changes and calculate the time required. The thermal conductivity is 0.69 W/m-K and the thermal diffusivity 5.85x10 -4 m 2 /h.
A long steel rod 0.305 m in diameter is initially at a temperature of 588K. It is immersed in an oil bath maintained at 311K. The surface convective coefficient is 125 W/m 2 -K. Calculate the temperature at the center of the rod after 1 h. The average physical properties of the steel are k=38 W/m-K and =0.0381m 2 /h
5.3-9 Temp. of Oranges on Trees During Freezing Weather
In orange-growing areas, the freezing of the oranges on the trees during cold nights is economically important. If the oranges are initially at a temperature of 21.1 o C, calculate the center temperature of the orange if exposed to air at –3.9 o C for 6 h. The oranges are 102 mm in diameter and the convective coefficient is estimated as 11.4W/m 2 -K. The thermal conductivity k is 0.431 W/m-K and =4.65x10 -4 m 2 /h. Neglect any latent heat effects.
A large piece of aluminum that can be considered a semi-infinite solid initially has a uniform temperature of 505.4K. The surface is suddenly exposed to an environment at 338.8K with a surface convection coefficient of 455W/m 2 -K. Calculate the time in hours for the temperature to reach 388.8 K at a depth of 25.4 mm. The average physical properties are =0.340m 2 /h and k=208W/m-K.